Average Error: 0.1 → 0.1
Time: 10.5s
Precision: 64
\[\left(1 - x\right) + y \cdot \sqrt{x}\]
↓
\[\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)\]
\left(1 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)
double f(double x, double y) {
        double r77798 = 1.0;
        double r77799 = x;
        double r77800 = r77798 - r77799;
        double r77801 = y;
        double r77802 = sqrt(r77799);
        double r77803 = r77801 * r77802;
        double r77804 = r77800 + r77803;
        return r77804;
}
↓
double f(double x, double y) {
        double r77805 = y;
        double r77806 = x;
        double r77807 = sqrt(r77806);
        double r77808 = 1.0;
        double r77809 = r77808 - r77806;
        double r77810 = fma(r77805, r77807, r77809);
        return r77810;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.1

    \[\left(1 - x\right) + y \cdot \sqrt{x}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)}\]

  3. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1 x) (* y (sqrt x))))